16) An integer is said to be square-free if it is notdivisable by the square of

Question

16) An integer is said to be square-free if it is notdivisable by the square of any integer greater than 1.         Prove thefollowing:    b) Every integer n > 1 is the product ofa square-free integer and a perfect square.         [Hint: Ifn=p1kp2k2***psksis the canonical factorization of n, then we writeki=2qi+ri where r = 0 or 1according as ki is even or odd. 16) An integer is said to be square-free if it is notdivisable by the square of any integer greater than 1.         Prove thefollowing:    b) Every integer n > 1 is the product ofa square-free integer and a perfect square.         [Hint: Ifn=p1kp2k2***psksis the canonical factorization of n, then we writeki=2qi+ri where r = 0 or 1according as ki is even or odd.

Explanation / Answer

Let n be any integer greater than 1 andn=p1kp2k2***psksbe the canonical prime factorization of n,
then we write ki=2qi+ri where r =0 or 1 according as ki is even or odd.

Then n =p12q1+r1p22q2+r2   ***ps2qs+rs
   = p12q1p22q2  ***ps2qs . p1r1p2r2   ***psrs
            = (p1q1p2q2   ***psqs )2 . (p1r1p2r2   ***psrs )
   = m.k
where m = (p1q1p2q2   ***psqs )2 .and k = (p1r1p2r2   ***psrs ) .
Note m is a perfect square. Also sinceri is either 1 or 0, no square of a prime dividesk.

So k is a square free integer.

   This proves the result

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